MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nmoofval Structured version   Visualization version   GIF version

Theorem nmoofval 29746
Description: The operator norm function. (Contributed by NM, 6-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmoofval.1 𝑋 = (BaseSetβ€˜π‘ˆ)
nmoofval.2 π‘Œ = (BaseSetβ€˜π‘Š)
nmoofval.3 𝐿 = (normCVβ€˜π‘ˆ)
nmoofval.4 𝑀 = (normCVβ€˜π‘Š)
nmoofval.6 𝑁 = (π‘ˆ normOpOLD π‘Š)
Assertion
Ref Expression
nmoofval ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ 𝑁 = (𝑑 ∈ (π‘Œ ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )))
Distinct variable groups:   π‘₯,𝑑,𝑧,π‘ˆ   𝑑,π‘Š,π‘₯,𝑧   𝑑,𝑋,𝑧   𝑑,π‘Œ,π‘₯   𝑑,𝐿   𝑑,𝑀
Allowed substitution hints:   𝐿(π‘₯,𝑧)   𝑀(π‘₯,𝑧)   𝑁(π‘₯,𝑧,𝑑)   𝑋(π‘₯)   π‘Œ(𝑧)

Proof of Theorem nmoofval
Dummy variables 𝑒 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmoofval.6 . 2 𝑁 = (π‘ˆ normOpOLD π‘Š)
2 fveq2 6843 . . . . . 6 (𝑒 = π‘ˆ β†’ (BaseSetβ€˜π‘’) = (BaseSetβ€˜π‘ˆ))
3 nmoofval.1 . . . . . 6 𝑋 = (BaseSetβ€˜π‘ˆ)
42, 3eqtr4di 2791 . . . . 5 (𝑒 = π‘ˆ β†’ (BaseSetβ€˜π‘’) = 𝑋)
54oveq2d 7374 . . . 4 (𝑒 = π‘ˆ β†’ ((BaseSetβ€˜π‘€) ↑m (BaseSetβ€˜π‘’)) = ((BaseSetβ€˜π‘€) ↑m 𝑋))
6 fveq2 6843 . . . . . . . . . . 11 (𝑒 = π‘ˆ β†’ (normCVβ€˜π‘’) = (normCVβ€˜π‘ˆ))
7 nmoofval.3 . . . . . . . . . . 11 𝐿 = (normCVβ€˜π‘ˆ)
86, 7eqtr4di 2791 . . . . . . . . . 10 (𝑒 = π‘ˆ β†’ (normCVβ€˜π‘’) = 𝐿)
98fveq1d 6845 . . . . . . . . 9 (𝑒 = π‘ˆ β†’ ((normCVβ€˜π‘’)β€˜π‘§) = (πΏβ€˜π‘§))
109breq1d 5116 . . . . . . . 8 (𝑒 = π‘ˆ β†’ (((normCVβ€˜π‘’)β€˜π‘§) ≀ 1 ↔ (πΏβ€˜π‘§) ≀ 1))
1110anbi1d 631 . . . . . . 7 (𝑒 = π‘ˆ β†’ ((((normCVβ€˜π‘’)β€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§))) ↔ ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))))
124, 11rexeqbidv 3319 . . . . . 6 (𝑒 = π‘ˆ β†’ (βˆƒπ‘§ ∈ (BaseSetβ€˜π‘’)(((normCVβ€˜π‘’)β€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§))) ↔ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))))
1312abbidv 2802 . . . . 5 (𝑒 = π‘ˆ β†’ {π‘₯ ∣ βˆƒπ‘§ ∈ (BaseSetβ€˜π‘’)(((normCVβ€˜π‘’)β€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))} = {π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))})
1413supeq1d 9387 . . . 4 (𝑒 = π‘ˆ β†’ sup({π‘₯ ∣ βˆƒπ‘§ ∈ (BaseSetβ€˜π‘’)(((normCVβ€˜π‘’)β€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))}, ℝ*, < ) = sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))}, ℝ*, < ))
155, 14mpteq12dv 5197 . . 3 (𝑒 = π‘ˆ β†’ (𝑑 ∈ ((BaseSetβ€˜π‘€) ↑m (BaseSetβ€˜π‘’)) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ (BaseSetβ€˜π‘’)(((normCVβ€˜π‘’)β€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )) = (𝑑 ∈ ((BaseSetβ€˜π‘€) ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )))
16 fveq2 6843 . . . . . 6 (𝑀 = π‘Š β†’ (BaseSetβ€˜π‘€) = (BaseSetβ€˜π‘Š))
17 nmoofval.2 . . . . . 6 π‘Œ = (BaseSetβ€˜π‘Š)
1816, 17eqtr4di 2791 . . . . 5 (𝑀 = π‘Š β†’ (BaseSetβ€˜π‘€) = π‘Œ)
1918oveq1d 7373 . . . 4 (𝑀 = π‘Š β†’ ((BaseSetβ€˜π‘€) ↑m 𝑋) = (π‘Œ ↑m 𝑋))
20 fveq2 6843 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ (normCVβ€˜π‘€) = (normCVβ€˜π‘Š))
21 nmoofval.4 . . . . . . . . . . 11 𝑀 = (normCVβ€˜π‘Š)
2220, 21eqtr4di 2791 . . . . . . . . . 10 (𝑀 = π‘Š β†’ (normCVβ€˜π‘€) = 𝑀)
2322fveq1d 6845 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)) = (π‘€β€˜(π‘‘β€˜π‘§)))
2423eqeq2d 2744 . . . . . . . 8 (𝑀 = π‘Š β†’ (π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)) ↔ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§))))
2524anbi2d 630 . . . . . . 7 (𝑀 = π‘Š β†’ (((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§))) ↔ ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))))
2625rexbidv 3172 . . . . . 6 (𝑀 = π‘Š β†’ (βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§))) ↔ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))))
2726abbidv 2802 . . . . 5 (𝑀 = π‘Š β†’ {π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))} = {π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))})
2827supeq1d 9387 . . . 4 (𝑀 = π‘Š β†’ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))}, ℝ*, < ) = sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))}, ℝ*, < ))
2919, 28mpteq12dv 5197 . . 3 (𝑀 = π‘Š β†’ (𝑑 ∈ ((BaseSetβ€˜π‘€) ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )) = (𝑑 ∈ (π‘Œ ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )))
30 df-nmoo 29729 . . 3 normOpOLD = (𝑒 ∈ NrmCVec, 𝑀 ∈ NrmCVec ↦ (𝑑 ∈ ((BaseSetβ€˜π‘€) ↑m (BaseSetβ€˜π‘’)) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ (BaseSetβ€˜π‘’)(((normCVβ€˜π‘’)β€˜π‘§) ≀ 1 ∧ π‘₯ = ((normCVβ€˜π‘€)β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )))
31 ovex 7391 . . . 4 (π‘Œ ↑m 𝑋) ∈ V
3231mptex 7174 . . 3 (𝑑 ∈ (π‘Œ ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )) ∈ V
3315, 29, 30, 32ovmpo 7516 . 2 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ (π‘ˆ normOpOLD π‘Š) = (𝑑 ∈ (π‘Œ ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )))
341, 33eqtrid 2785 1 ((π‘ˆ ∈ NrmCVec ∧ π‘Š ∈ NrmCVec) β†’ 𝑁 = (𝑑 ∈ (π‘Œ ↑m 𝑋) ↦ sup({π‘₯ ∣ βˆƒπ‘§ ∈ 𝑋 ((πΏβ€˜π‘§) ≀ 1 ∧ π‘₯ = (π‘€β€˜(π‘‘β€˜π‘§)))}, ℝ*, < )))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆƒwrex 3070   class class class wbr 5106   ↦ cmpt 5189  β€˜cfv 6497  (class class class)co 7358   ↑m cmap 8768  supcsup 9381  1c1 11057  β„*cxr 11193   < clt 11194   ≀ cle 11195  NrmCVeccnv 29568  BaseSetcba 29570  normCVcnmcv 29574   normOpOLD cnmoo 29725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-sup 9383  df-nmoo 29729
This theorem is referenced by:  nmooval  29747  hhnmoi  30885
  Copyright terms: Public domain W3C validator